cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A027879 a(n) = Product_{i=1..n} (11^i - 1).

Original entry on oeis.org

1, 10, 1200, 1596000, 23365440000, 3763004112000000, 6666387564654720000000, 129909027758312519942400000000, 27847153692160782464830528512000000000, 65662131721505488121539650946349537280000000000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

It appears that the number of trailing zeros in a(n) is A191610(n). - Robert Israel, Nov 24 2015

Links

Crossrefs

Cf. A005329 (q=2), A027871 (q=3), A027637 (q=4), A027872 (q=5), A027873 (q=6), A027875 (q=7), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027880 (q=12).
Cf. A132267, A191610.

Programs

  • Magma
    [1] cat [&*[11^k-1: k in [1..n]]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015
  • Maple
    seq(mul(11^i-1,i=1..n),n=0..20; # Robert Israel, Nov 24 2015
  • Mathematica
    FoldList[Times,1,11^Range[10]-1] (* Harvey P. Dale, Aug 13 2013 *)
Abs@QPochhammer[11, 11, Range[0, 40]] (* G. C. Greubel, Nov 24 2015 *)
  • PARI
    a(n)=prod(i=1,n,11^i-1) \\ Anders Hellström, Nov 21 2015

Formula

10^n|a(n) for n>=0; 12*(10)^(n)|a(n) n>=2. - G. C. Greubel, Nov 21 2015
a(n) ~ c * 11^(n*(n+1)/2), where c = Product_{k>=1} (1-1/11^k) = 0.900832706809715279949862694760647744762491192216... . - Vaclav Kotesovec, Nov 21 2015
E.g.f. E(x) satisfies E'(x) = 11 E(11 x) - E(x). - Robert Israel, Nov 24 2015
Equals 11^(binomial(n+1,2))*(1/11;1/11){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 24 2015
G.f.: Sum_{n>=0} 11^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 11^k*x). - Ilya Gutkovskiy, May 22 2017
Sum_{n>=0} (-1)^n/a(n) = A132267. - Amiram Eldar, May 07 2023

A132263 Product{0<=k<=floor(log_11(n)), floor(n/11^k)}, n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 275, 280, 285, 290, 295, 300, 305, 310
Offset: 1

Views

Author

Hieronymus Fischer, Aug 20 2007

Keywords

Comments

If n is written in base-11 as n=d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product d(m)d(m-1)d(m-2)...d(2)d(1)d(0)*d(m)d(m-1)d(m-2)...d(2)d(1)*d(m)d(m-1)d(m-2)...d(2)*...*d(m)d(m-1)d(m-2)*d(m)d(m-1)*d(m).

Examples

			a(50)=floor(50/11^0)*floor(50/11^1)=50*4=200; a(63)=315 since 63=58(base-11) and so a(63)=58*5(base-11)=63*5=315.

Crossrefs

Cf. A048651, A132019-A132026, A132265, A132267, A000217.
For formulas regarding a general parameter p (i.e. terms floor(n/p^k)) see A132264.
For the product of terms floor(n/p^k) for p=2 to p=12 see A098844(p=2), A132027(p=3)-A132033(p=9), A067080(p=10), A132264(p=12).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

Formula

Recurrence: a(n)=n*a(floor(n/11)); a(n*11^m)=n^m*11^(m(m+1)/2)*a(n).
a(k*11^m)=k^(m+1)*11^(m(m+1)/2), for 0
Asymptotic behavior: a(n)=O(n^((1+log_11(n))/2)); this follows from the inequalities below.
a(n)<=b(n), where b(n)=n^(1+floor(log_11(n)))/p^((1+floor(log_11(n)))*floor(log_11(n))/2); equality holds for n=k*11^m, 0=0. b(n) can also be written n^(1+floor(log_11(n)))/11^A000217(floor(log_11(n))).
Also: a(n)<=3^((1-log_11(3))/2)*n^((1+log_11(n))/2)=1.346673852...^((1-log_11(3))/2)*11^A000217(log_11(n)), equality holds for n=3*11^m, m>=0.
a(n)>c*b(n), where c=0.4751041275076031053975644472... (see constant A132265).
Also: a(n)>c*(sqrt(2)/2^log_11(sqrt(2)))*n^((1+log_11(n))/2)=0.607848303...*11^00217(log_11(n)).
lim inf a(n)/b(n)=0.4751041275076031053975644472..., for n-->oo.
lim sup a(n)/b(n)=1, for n-->oo.
lim inf a(n)/n^((1+log_p(n))/2)=0.4751041275076031...*sqrt(2)/2^log_11(sqrt(2)), for n-->oo.
lim sup a(n)/n^((1+log_p(n))/2)=sqrt(3)/3^log_11(sqrt(3))=1.346673852..., for n-->oo.
lim inf a(n)/a(n+1)=0.4751041275076031053975644472... for n-->oo (see constant A132265).

A132264 Product{0<=k<=floor(log_12(n)), floor(n/12^k)}, n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236, 300, 305, 310, 315
Offset: 1

Views

Author

Hieronymus Fischer, Aug 20 2007

Keywords

Comments

If n is written in base-12 as n=d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product d(m)d(m-1)d(m-2)...d(2)d(1)d(0)*d(m)d(m-1)d(m-2)...d(2)d(1)*d(m)d(m-1)d(m-2)...d(2)*...*d(m)d(m-1)d(m-2)*d(m)d(m-1)*d(m).

Examples

			a(50)=floor(50/12^0)*floor(50/12^1)=50*4=200.
a(65)=325 since 65=55(base-12) and so a(65)=55*5(base-12)=65*5=325.

Crossrefs

Cf. A048651, A132019-A132026, A132265, A132267, A000217, A002378.
For the product of terms floor(n/p^k) for p=2 to p=11 see A098844(p=2), A132027(p=3)-A132033(p=9), A067080(p=10), A132263(p=11).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

Formula

The following formulas are given for a general parameter p considering the product of terms floor(n/p^k) for 0<=k<=floor(log_p(n)), where p=12 for this sequence.
Recurrence: a(n)=n*a(floor(n/p)); a(n*p^m)=n^m*p^(m(m+1)/2)*a(n).
a(k*p^m)=k^(m+1)*p^(m(m+1)/2), for 0
Asymptotic behavior: a(n)=O(n^((1+log_p(n))/2)); this follows from the inequalities below.
a(n)<=b(n), where b(n)=n^(1+floor(log_p(n)))/p^((1+floor(log_p(n)))*floor(log_p(n))/2); equality holds for n=k*p^m, 0=0. b(n) can also be written n^(1+floor(log_p(n)))/p^A000217(floor(log_p(n))).
Also: a(n)<=q^((1-log_p(q))/2)*n^((1+log_p(n))/2)=q^((1-log_p(q))/2)*p^A000217(log_p(n)), equality holds for n=q*p^m, m>=0, where q=floor(sqrt(p)+1/2). Also, equality holds for n=(q+1)*p^m, provided p is a A002378-number (in this case we have p=q*(q+1) and so q^((1-log_p(q))/2)=(q+1)^((1-log_p(q+1))/2)).
a(n)>c*b(n), where c=product{k>0, 1-1/(2*p^k)}=0.47735217025489380... (for p=12 see constant A132265).
Also: a(n)>c*(sqrt(2)/2^log_p(sqrt(2)))*n^((1+log_p(n))/2)=0.612870619...*p^A000217(log_p(n)), (p=12).
lim inf a(n)/b(n)=product{k>0, 1-1/(2*p^k)}=0.47735217025489380198334286365820..., for n-->oo (for p=12 see constant A132265).
lim sup a(n)/b(n)=1, for n-->oo.
lim inf a(n)/n^((1+log_p(n))/2)=(sqrt(2)/2^log_p(sqrt(2)))*product{k>0, 1-1/(2*p^k)}=0.612870619..., for n-->oo, (p=12).
lim sup a(n)/n^((1+log_p(n))/2)=sqrt(q)/q^log_p(sqrt(q))=1.358593737..., for n-->oo, (p=12, q=round(sqrt(p))=3).
lim inf a(n)/a(n+1)=product{k>0, 1-1/(2*p^k)}=0.47735217025489380... for n-->oo (for p=12 see constant A132265).

A015011 q-factorial numbers for q=11.

Original entry on oeis.org

1, 1, 12, 1596, 2336544, 37630041120, 6666387564654720, 12990902775831251994240, 278471536921607824648305285120, 65662131721505488121539650946349537280, 170310659060181679663863033233125976844488908800, 4859161865915056755501262525796512204608930674134393036800
Offset: 0

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A015001, A015002, A015004, A015005, A015006, A015007, A015008, A015009.
Column q=11 of A069777.
Cf. A016123, A132267.

Programs

  • Magma
    [n le 1 select 1 else (11^n-1)*Self(n-1)/10: n in [1..15]]; // Vincenzo Librandi, Oct 26 2012
  • Mathematica
    RecurrenceTable[{a[1]==1, a[n]==((11^n - 1) * a[n-1])/10}, a, {n, 15}] (* Vincenzo Librandi, Oct 26 2012 *)
Table[QFactorial[n, 11], {n, 11}] (* Bruno Berselli, Aug 14 2013 *)

Formula

a(n) = Product_{k=1..n} (11^k - 1) / (11 - 1).
a(0) = 1, a(n) = (11^n - 1)*a(n-1)/10. - Vincenzo Librandi, Oct 26 2012
From Amiram Eldar, Jul 05 2025: (Start)
a(n) = Product_{k=1..n} A016123(k-1).
a(n) ~ c * 11^(n*(n+1)/2)/10^n, where c = A132267. (End)

Extensions

a(0)=1 prepended by Alois P. Heinz, Sep 08 2021

A132268 Decimal expansion of Product_{k>0} (1-1/12^k).

Original entry on oeis.org

9, 0, 9, 7, 2, 6, 2, 6, 8, 9, 0, 5, 9, 9, 4, 8, 8, 8, 6, 3, 6, 3, 6, 2, 0, 4, 6, 9, 7, 7, 0, 8, 0, 2, 4, 9, 1, 2, 0, 7, 9, 1, 6, 9, 1, 9, 4, 1, 0, 1, 4, 2, 7, 4, 3, 2, 6, 1, 5, 4, 4, 4, 1, 2, 8, 6, 9, 0, 2, 4, 5, 7, 6, 6, 1, 9, 5, 4, 1, 6, 2, 0, 2, 6, 0, 0, 0, 5, 5, 3, 8, 8, 8, 8, 1, 0, 8, 5, 1, 4, 8, 3, 9, 7, 1, 9
Offset: 0

Views

Author

Hieronymus Fischer, Aug 20 2007

Keywords

Examples

			0.9097262689059948886363620469770...

Links

Crossrefs

Cf. A027880, A048651, A100220, A132019-A132026, A132034-A132038, A132265-A132267, A132323-A132326, A000203.

Programs

  • Mathematica
    digits = 106; NProduct[1-1/12^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
N[QPochhammer[1/12,1/12]] (* G. C. Greubel, Dec 05 2015 *)
  • PARI
    prodinf(k=1, (1-1/12^k)) \\ Michel Marcus, Dec 05 2015

Formula

Equals exp(-Sum_{n>0} sigma_1(n)/(n*12^n)) = exp(-Sum_{n>0} A000203(n)/(n*12^n)).
Equals (1/12; 1/12){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 05 2015
From Amiram Eldar, May 09 2023: (Start)
Equals sqrt(2*Pi/log(12)) * exp(log(12)/24 - Pi^2/(6*log(12))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(12))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027880(n). (End)

A028673 Galois numbers for p=11; order of group AGL(n,11).

Original entry on oeis.org

1, 110, 1597200, 2827411356000, 606039338269189440000, 15719002038355567350156912000000, 49332934203739383347738694321468865920000000, 18734172592683683919731709047397914403374452828934400000000, 860830165835516295815608223447061872667128267986790628055380728832000000000
Offset: 0

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A028365, A028665, A028667, A028669, A028675, A028679, A028681, A028685.
Cf. A132267.

Programs

  • Mathematica
    FoldList[ #1*11^#2 (11^#2-1)&, 1, Range[ 20 ] ]
a[n_] := 11^n * Product[11^n - 11^k, {k, 0, n-1}]; Array[a, 10, 0] (* Amiram Eldar, Jul 12 2025 *)
  • PARI
    a(n) = 11^n * prod(k = 0, n-1, 11^n - 11^k); \\ Amiram Eldar, Jul 12 2025

Formula

a(n) = 11^n * Product_{k=0..n-1} (11^n - 11^k).
a(n) ~ c * 11^(n^2+n), where c = A132267. - Amiram Eldar, Jul 12 2025

A052498 Number of nonsingular n X n matrices over GF(11).

Original entry on oeis.org

1, 10, 13200, 2124276000, 41393302251840000, 97602635428252959312000000, 27847155251069188894843979022720000000, 961359275427083998992553051820498439890246400000000
Offset: 0

Views

Author

Vladeta Jovovic, Mar 16 2000

Keywords

Links

Crossrefs

Cf. A027879, A110195.
Cf. A002884, A053290, A053291, A053292, A053293, A052496, A052497, A132267.

Programs

  • Magma
    [1] cat [&*[(11^n - 11^k): k in [0..n-1]]: n in [1..10]]; // Bruno Berselli, Jan 28 2013
  • Mathematica
    Table[Product[11^n - 11^k, {k, 0, n-1}], {n, 0, 10}] (* Vincenzo Librandi, Jan 28 2013 *)
  • PARI
    {a(n) = prod(j=0,n-1, 11^n - 11^j)}; \\ G. C. Greubel, May 14 2019
  • Sage
    [product(11^n - 11^j for j in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 14 2019

Formula

a(n) = (11^n - 1)*(11^n - 11)*...*(11^n - 11^(n-1)).
a(n) = A110195(n)*A027879(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 11^(n^2), where c = A132267. - Amiram Eldar, Jul 06 2025
Showing 1-7 of 7 results.

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